Use Y1 to Evaluate a Regression Prediction
In “Making Predictions Using the Equation,” you substituted a predictor value into a regression equation by hand. A TI-84 can evaluate the same equation for you. The key is to store the regression equation in the calculator’s \(Y1\) function when you run LinReg(a+bx). Then, on the home screen, enter \(Y1(x)\) with the predictor value in parentheses.
This is a convenient way to get predictions from a fitted line, especially when you need several predictions. It does not change what the regression equation means: \(Y1(x)\) returns the line’s predicted response, \(\hat{y}\), for the specified predictor value \(x\). It does not return an observed response or guarantee what will happen in an individual case.
Store the Regression Equation in Y1
We will use a fictional greenhouse example. A student records the amount of supplemental light seedlings receive each day and the seedlings’ growth over the following week. The predictor \(x\) is supplemental light in hours per day, and the response \(y\) is growth in millimeters. The paired observations are entered in matching positions in \(L1\) and \(L2\), as in “Entering Data and Running LinReg on a TI-84.”
| Supplemental light, \(x\) (hours per day) | Growth, \(y\) (millimeters) |
|---|---|
| 1 | 13 |
| 2 | 20 |
| 3 | 21 |
| 4 | 26 |
| 5 | 25 |
On a TI-84, choose the linear regression command LinReg(a+bx) from the STAT CALC menu. Enter the predictor list first and the response list second. To store the equation as part of this command, add \(Y1\) as the regression equation destination. On many TI-84 models, the home-screen command is:
Use the calculator’s VARS menu, then Y-VARS and Function, to select \(Y1\). The precise menu navigation can differ slightly by TI-84 model, but the essential order is predictor list, response list, and then the destination function. When you press ENTER, the calculator reports the regression coefficients and stores the fitted equation in \(Y1\). Storing the equation replaces any equation that was already in \(Y1\).
For these data, the calculator gives an intercept \(a=12\) and a slope \(b=3\). Thus, the equation stored in \(Y1\) is:
Here, \(Y1\) is the calculator’s name for the function represented by the regression equation. The coefficients have the same meanings as in “Reading the Equation of a Regression Line”: the intercept is the predicted response at \(x=0\), and the slope is the change in predicted response for a one-unit increase in \(x\). In context, the fitted line predicts 3 additional millimeters of growth for each additional hour of supplemental light per day. The prediction’s units are millimeters.
Evaluate Y1 on the Home Screen
After storing the equation, return to the home screen. Choose \(Y1\) from the VARS, Y-VARS, Function menu, or enter the function name directly if your calculator allows it. Put the chosen predictor value in parentheses immediately after \(Y1\). For example, to evaluate the fitted equation at \(x=4\), enter \(Y1(4)\) and press ENTER.
The parentheses are important: they show which value is being supplied as the input to the function. The calculator substitutes that value for \(x\) in the stored regression equation and evaluates the result. You do not need to re-enter the coefficients each time, and you do not need to use the graph screen to get a numerical prediction.
Use LinReg(a+bx) with the predictor list first, the response list second, and \(Y1\) as the storage destination.
Confirm that the calculator’s coefficients give the expected form, \(\hat{y}=a+bx\).
On the home screen, enter \(Y1(x)\), placing the requested value inside parentheses.
Report the output as a predicted response, name the setting, and use the response variable’s units.
Worked Examples
Worked Example: Store the Line and Predict at 4 Hours
Use the greenhouse data in \(L1\) and \(L2\). Store the linear regression equation in \(Y1\), then find the predicted growth for a seedling receiving 4 hours of supplemental light per day.
Run LinReg(a+bx) with \(L1\) as the predictor list, \(L2\) as the response list, and \(Y1\) as the storage destination. The calculator reports \(a=12\) and \(b=3\), so the stored equation is \(\hat{y}=12+3x\). Enter \(Y1(4)\) on the home screen. The calculator evaluates:
For a seedling receiving 4 hours of supplemental light per day, the fitted line predicts growth of 24 millimeters over the following week. The answer is in millimeters because growth is the response variable. The calculator has evaluated the line; 24 millimeters is not necessarily the actual growth of a particular seedling.
Worked Example: Use the Same Stored Equation for a Decimal Input
The equation is still stored in \(Y1\). Find the predicted growth for a seedling receiving 2.5 hours of supplemental light per day. Enter \(Y1(2.5)\) on the home screen.
The regression line predicts 19.5 millimeters of growth for 2.5 hours of supplemental light per day. The decimal input is valid: \(Y1\) can evaluate the equation at decimal values as well as whole numbers. The prediction is a value on the fitted line, not a claim that a measured seedling’s growth will be exactly 19.5 millimeters.
A useful check is to evaluate the equation directly: \(12+3(2.5)=12+7.5=19.5\). If your calculator gives a different value, confirm that you entered \(2.5\) inside the parentheses and that \(Y1\) contains the intended regression equation.
Worked Example: Make Several Predictions Without Repeating the Regression
Suppose you want predictions for supplemental-light amounts of 1.5 and 4.5 hours per day. Since the line is already stored in \(Y1\), enter each input separately on the home screen.
The fitted line predicts 16.5 millimeters of growth for 1.5 hours of supplemental light per day and 25.5 millimeters for 4.5 hours per day. Both calculations use the same stored equation. There is no need to run LinReg again unless the data or the regression equation you want to use has changed.
Both inputs are between the smallest and largest observed values, 1 and 5 hours per day. They therefore use the line within the observed predictor range. As discussed in “Making Predictions Using the Equation,” a value outside the observed range would be extrapolation; the calculator could still return a number, but that alone would not establish that the prediction is reliable.
What to Check When a Prediction Looks Wrong
The calculator can only evaluate the function that is currently stored in \(Y1\). If a prediction is unexpected, first check that you stored the regression equation and not a different equation. You can view the function by opening the \(Y=\) screen. For this example, \(Y1\) should display the fitted line corresponding to \(12+3x\).
Next, check the order of the lists used in LinReg(a+bx). The predictor values must be first and the response values second. Reversing the lists fits a different line, because it switches which variable is used to predict the other. Also confirm that each predictor is paired with its matching response in the same list position, as described in “Entering Data and Running LinReg on a TI-84.”
Finally, check the input and its interpretation. \(Y1(2.5)\) means evaluate the stored equation at \(x=2.5\); it does not mean multiply \(Y1\) by 2.5. The output is measured in response units. If \(x\) is in hours per day and \(y\) is growth in millimeters, the output is millimeters—not hours.
Common Mistakes and AP Exam Tips
- Running the regression without storing the equation. The calculator may display coefficients without saving the line in \(Y1\). Include \(Y1\) as the regression equation destination when you run LinReg(a+bx).
- Using the lists in the wrong order. Enter the explanatory-variable list first and the response-variable list second. The regression equation predicts the response from the predictor.
- Forgetting parentheses. Enter \(Y1(4)\), not just \(Y1\), when you want the predicted response at \(x=4\).
- Using an old or unintended equation. A prior equation in \(Y1\) may remain if you did not store the new one. Check the \(Y=\) screen and verify the coefficients before interpreting results.
- Giving the predictor’s units to the prediction. The home-screen output is a predicted response. State its response units and explain what it predicts in context.
- Treating the output as an observed or guaranteed value. Say that the regression line predicts the response. Individual observed responses can differ from the line’s prediction.
- Assuming a calculator result validates an extrapolation. The calculator evaluates the equation at any numerical input, but predictions beyond the observed range may be unreliable.
For full-credit communication, do not stop at the calculator display. State the predictor value, identify the predicted response, include its units, and use wording such as “the regression line predicts.” For example: “For 2.5 hours of supplemental light per day, the fitted line predicts 19.5 millimeters of seedling growth over the following week.”
Check Your Understanding
Use the greenhouse equation \(\hat{y}=12+3x\), where \(x\) is supplemental light in hours per day and \(\hat{y}\) is predicted growth in millimeters.
- What home-screen expression would you enter to find the predicted growth for 3 hours of supplemental light per day?
- Evaluate \(Y1(3)\), showing the substitution and the result with units.
- What does the calculator need to do with the LinReg(a+bx) command for \(Y1(x)\) to evaluate the intended regression equation?
- If the home screen displays 19.5 after entering \(Y1(2.5)\), what does that number represent in context?
- Why does a numerical output from \(Y1(7)\) not by itself show that the prediction is reliable?